Systems of Conservation Laws

Abstract

The motion of a compressible fluid with negligible viscosity (gas dynamics) or the propagation of waves in shallow waters are typical phenomena leading to systems of first order conservation laws. This chapter constitutes an introduction to the basic concepts in this important area of nonlinear PDEs, still lacking of a completely satisfactory theory. Let us introduce the following notations: $$ \begin{array}{*{20}{c}} {u:{\mathbb{R}^n} \times \left[ {0,T} \right] \to {\mathbb{R}^m},}&{F:U \subseteq {\mathbb{R}^m} \to {\mathcal{M}_{m,n}}} \end{array} $$ where M m,n is the set of m × n matrices. In this context, u is a state variable, and we write it as a column vector or as a point in ℝ m . If Ω ⊂ ℝ n is a bounded domain, the equation $$ \frac{d}{{dt}}\int_\Omega {udx = } - \int_{\partial \Omega } {F\left( u \right) \cdot vd\sigma } $$ (11.1) expresses the balance between the rate of variation of \( \int_\Omega {udx} \) and the inward flux of u through ∂Ω, governed by the flux function F. As usual, v denotes the outward unit normal to ∂Ω. Using Gauss formula, we rewrite (11.1) in the form $$ \int_\Omega {\left[ {{u_t} + divF\left( u \right)} \right]dx = 0.} $$ (11.2a)